D40:1C4 - GroupNames

G = D₄₀⋊₁C₄ order 320 = 2⁶·5

1^st semidirect product of D₄₀ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄₀⋊₁C₄, Q₁₆⋊₂F₅, D₁₀.₃D₈, Dic₅.₁₇SD₁₆, C₈.₃(C₂×F₅), C₄₀.₃(C₂×C₄), C₄₀⋊C₄⋊₂C₂, (C₅×Q₁₆)⋊₁C₄, C₅⋊₂(D₈⋊₂C₄), (C₄×D₅).₂₄D₄, C₅⋊₂C₈.₁₅D₄, C₈.F₅⋊₃C₂, C₄.₅(C₂²⋊F₅), Q₈.D₁₀.₂C₂, C₂₀.₅(C₂²⋊C₄), (C₈×D₅).₁₁C₂², C₁₀.₉(D₄⋊C₄), C₂.₁₀(D₂₀⋊C₄), SmallGroup(320,245)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₀ — D₄₀⋊₁C₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄×D₅ — C₈×D₅ — C₄₀⋊C₄ — D₄₀⋊₁C₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄₀ — D₄₀⋊₁C₄

Upper central

C₁ — C₂ — C₄ — C₈ — Q₁₆

Generators and relations for D₄₀⋊₁C₄
G = < a,b,c | a⁴⁰=b²=c⁴=1, bab=a^-1, cac^-1=a³, cbc^-1=a³⁷b >

Subgroups: 346 in 58 conjugacy classes, 20 normal (all characteristic)
C₁, C₂, C₂^[×2], C₄, C₄^[×3], C₂²^[×2], C₅, C₈, C₈, C₂×C₄^[×3], D₄^[×2], Q₈, D₅^[×2], C₁₀, C₁₆, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₄.Q₈, M₅(2), C₄○D₈, C₅⋊₂C₈, C₄₀, C₄×D₅, C₄×D₅, D₂₀^[×2], C₅×Q₈, C₂×F₅, D₈⋊₂C₄, C₅⋊C₁₆, C₈×D₅, D₄₀, Q₈⋊D₅, C₅×Q₁₆, C₄⋊F₅, Q₈⋊₂D₅, C₈.F₅, C₄₀⋊C₄, Q₈.D₁₀, D₄₀⋊₁C₄
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], C₂²⋊C₄, D₈, SD₁₆, F₅, D₄⋊C₄, C₂×F₅, D₈⋊₂C₄, C₂²⋊F₅, D₂₀⋊C₄, D₄₀⋊₁C₄

Character table of D₄₀⋊₁C₄

class	1	2A	2B	2C	4A	4B	4C	4D	4E	5	8A	8B	8C	10	16A	16B	16C	16D	20A	20B	20C	40A	40B
size	1	1	10	40	2	8	10	40	40	4	4	10	10	4	20	20	20	20	8	16	16	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	-1	-1	1	1	linear of order 2
ρ₄	1	1	1	-1	1	-1	1	-1	-1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	linear of order 2
ρ₅	1	1	-1	1	1	-1	-1	i	-i	1	1	-1	-1	1	-i	i	i	-i	1	-1	-1	1	1	linear of order 4
ρ₆	1	1	-1	-1	1	1	-1	i	-i	1	1	-1	-1	1	i	-i	-i	i	1	1	1	1	1	linear of order 4
ρ₇	1	1	-1	1	1	-1	-1	-i	i	1	1	-1	-1	1	i	-i	-i	i	1	-1	-1	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	1	-1	-i	i	1	1	-1	-1	1	-i	i	i	-i	1	1	1	1	1	linear of order 4
ρ₉	2	2	2	0	2	0	2	0	0	2	-2	-2	-2	2	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	-2	0	2	0	-2	0	0	2	-2	2	2	2	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	2	2	0	-2	0	-2	0	0	2	0	0	0	2	√2	-√2	√2	-√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₂	2	2	2	0	-2	0	-2	0	0	2	0	0	0	2	-√2	√2	-√2	√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₃	2	2	-2	0	-2	0	2	0	0	2	0	0	0	2	-√-2	-√-2	√-2	√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₄	2	2	-2	0	-2	0	2	0	0	2	0	0	0	2	√-2	√-2	-√-2	-√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₅	4	4	0	0	4	-4	0	0	0	-1	4	0	0	-1	0	0	0	0	-1	1	1	-1	-1	orthogonal lifted from C₂×F₅
ρ₁₆	4	4	0	0	4	4	0	0	0	-1	4	0	0	-1	0	0	0	0	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₁₇	4	4	0	0	4	0	0	0	0	-1	-4	0	0	-1	0	0	0	0	-1	-√5	√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₈	4	4	0	0	4	0	0	0	0	-1	-4	0	0	-1	0	0	0	0	-1	√5	-√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₉	4	-4	0	0	0	0	0	0	0	4	0	-2√-2	2√-2	-4	0	0	0	0	0	0	0	0	0	complex lifted from D₈⋊₂C₄
ρ₂₀	4	-4	0	0	0	0	0	0	0	4	0	2√-2	-2√-2	-4	0	0	0	0	0	0	0	0	0	complex lifted from D₈⋊₂C₄
ρ₂₁	8	8	0	0	-8	0	0	0	0	-2	0	0	0	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₂₀⋊C₄, Schur index 2
ρ₂₂	8	-8	0	0	0	0	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	0	-√10	√10	orthogonal faithful, Schur index 2
ρ₂₃	8	-8	0	0	0	0	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	0	√10	-√10	orthogonal faithful, Schur index 2

Smallest permutation representation of D₄₀⋊₁C₄
►On 80 points

Generators in S₈₀

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)

(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)(41 74 45 62)(42 61 54 65)(43 48 63 68)(44 75 72 71)(46 49 50 77)(47 76 59 80)(51 64 55 52)(53 78 73 58)(56 79 60 67)(57 66 69 70)

G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53)], [(2,28,10,4),(3,15,19,7),(5,29,37,13),(6,16),(8,30,24,22),(9,17,33,25),(11,31),(12,18,20,34),(14,32,38,40),(23,35,39,27),(26,36),(41,74,45,62),(42,61,54,65),(43,48,63,68),(44,75,72,71),(46,49,50,77),(47,76,59,80),(51,64,55,52),(53,78,73,58),(56,79,60,67),(57,66,69,70)])

Matrix representation of D₄₀⋊₁C₄ ►in GL₈(𝔽₂₄₁)

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

G:=sub<GL(8,GF(241))| [0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,222,222,192,13,0,0,0,0,19,222,114,129,0,0,0,0,0,0,38,137,0,0,0,0,0,0,95,0],[234,117,234,0,0,0,0,0,124,0,7,7,0,0,0,0,117,124,124,117,0,0,0,0,7,7,0,124,0,0,0,0,0,0,0,0,192,192,222,135,0,0,0,0,114,114,19,59,0,0,0,0,38,0,0,116,0,0,0,0,95,95,0,176],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,49,206,0,0,0,0,0,240,127,194,0,0,0,0,0,0,0,137,0,0,0,0,0,0,146,0] >;

D₄₀⋊₁C₄ in GAP, Magma, Sage, TeX

D_{40}\rtimes_1C_4

% in TeX

G:=Group("D40:1C4");

// GroupNames label

G:=SmallGroup(320,245);

// by ID

G=gap.SmallGroup(320,245);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,675,794,80,1684,851,102,6278,3156]);

// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;

// generators/relations

Export

Character table of D₄₀⋊₁C₄ in TeX

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

G = D40⋊1C4 order 320 = 26·5

1st semidirect product of D40 and C4 acting faithfully

G = D₄₀⋊₁C₄ order 320 = 2⁶·5

1^st semidirect product of D₄₀ and C₄ acting faithfully

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0